Which of the following are correct expressions for the torque acting on a body?
A. $\vec\tau = \vec r\times\vec L$
B. $\vec\tau = \dfrac{d}{dt}(\vec r\times\vec p)$
C. $\vec\tau = \vec r\times\dfrac{d\vec p}{dt}$
D. $\vec\tau = I\vec\alpha$
E. $\vec\tau = \vec r\times\vec F$
($\vec r$ = position vector; $\vec p$ = linear momentum; $\vec L$ = angular momentum; $\vec\alpha$ = angular acceleration; $I$ = moment of inertia; $\vec F$ = force; $t$ = time)
Choose the correct answer from the options given below:
Answer: (C) B, C, D and E only
- E. $\vec\tau = \vec r\times\vec F$ is the definition ✔
- C. $\vec F = \dfrac{d\vec p}{dt}$, so $\vec r\times\dfrac{d\vec p}{dt}$ is the same thing ✔
- B. $\dfrac{d}{dt}(\vec r\times\vec p) = \vec v\times m\vec v + \vec r\times\dfrac{d\vec p}{dt} = \vec r\times\vec F$, i.e. $\vec\tau = \dfrac{d\vec L}{dt}$ ✔
- D. For rotation about a fixed axis, $\tau = I\alpha$ ✔
- A. $\vec r\times\vec L$ does not even have the dimensions of torque ✘
Answer: B, C, D and E only.
Solution by Sreeraj P, M.Sc Physics